A service station has both self-service and full-service islands. On each island

Question

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation. (Give answers accurate to 3 decimal places.)

(a) Given that X = 1, determine the conditional pmf of Y—i.e., pY|X (0|1), pY|X (1|1), pY|X (2|1).

(b) Given that two hoses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island?

y p(x,y) 0 1 2 x 0 0.10 0.04 0.02 1 0.10 0.20 0.04 2 0.05 0.12 0.33

Explanation / Answer

a)

As P(x = 1) = 0.10+0.20+0.04 = 0.34, then we divide the entries in row x = 1 by 0.34 to get

b)

As P(x = 2) = 0.05+0.12+0.33 = 0.50, then we divide the entries in row x = 2 by 0.50 to get

c)

Hence, from b)

P(y<=1|=2) = 0.1 + 0.24 = 0.34 [ANSWER]

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d)

As P(y = 2) = 0.02+0.04+0.33 = 0.39, then we divide the entries in y = 2 column by 0.39 to get

Y p(Y|X=1) 0 0.29411765 1 0.58823529 2 0.11764706

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